The main result of [1]} is that the existence part of Question \(W^{thin}_{\mathbb {ADS}^3}\)  has a positive answer for metrics of constant curvature \(K\le -1\) : given \(K\le -1\)  and a quasi-symmetric homeomorphism \(\sigma :{\mathbb {RP}}^1\rightarrow {\mathbb {RP}}^1\) , there exists a convex domain \(\Omega \subset \mathbb {ADS}^3\)  with boundary at infinity a quasi-circle, such that the induced metrics on the two boundary components of \(\Omega \)  have constant curvature \(K\) , and the gluing map at infinity between them is equal to \(\sigma \) .
